of the factor 2 effect (B2) remains the same regardless of what other factors are included in the model. The net effect of the above two properties is that a factor effect can be computed once, and that value will hold for any linear model involving that term regardless of how simple or complicated the model is, provided that the design is orthogonal. This process greatly simplifies the model-building process because the need to recalculate all of the model coefficients for each new model is eliminated. Why is 1/2 the appropriate multiplicative term in these orthogonal models? Given the computational simplicity of orthogonal designs, why then is 1/2 the appropriate multiplicative constant? Why not 1/3, 1/4, etc.? To answer this, we revisit our specified desire that when we view the final fitted model and look at the coefficient associated with X2, say, we want the value of the coefficient B2 to reflect identically the expected total change Y in the response Y as we proceed from the "-" setting of X2 to the "+" setting of X2 (that is, we would like the estimated coefficient B2 to be identical to the estimated effect E2 for factor X2). Thus in glancing at the final model with this form, the coefficients B of the model will immediately reflect not only the relative importance of the coefficients, but will also reflect (absolutely) the effect of the associated term (main effect or interaction) on the response. In general, the least squares estimate of a coefficient in a linear model will yield a coefficient that is essentially a slope: = (change in response)/(change in factor levels) associated with a given factor X. Thus in order to achieve the desired interpretation of the coefficients B as being the raw change in the Y ( Y), we must account for and remove the change in X ( X). What is the X? In our design descriptions, we have chosen the notation of Box, Hunter and Hunter (1978) and set each (coded) factor to levels of "-" and "+". This "-" and "+" is a shorthand notation for -1 and +1. The advantage of this notation is that 2-factor interactions (and any higher-order interactions) also uniformly take on the closed values of -1 and +1, since -1*-1 = +1 -1*+1 = -1 +1*-1 = -1 +1*+1 = +1 and hence the set of values that the 2-factor interactions (and all interactions) take on are in the closed set {-1,+1}. This -1 and +1 notation is superior in its consistency to the (1,2) notation of Taguchi 5.5.9.9.6. Motivation: Why is the 1/2 in the Model? http://www.itl.nist.gov/div898/handbook/pri/section5/pri5996.htm (3 of 4) [5/1/2006 10:31:36 AM] in which the interaction, say X1*X2, would take on the values 1*1 = 1 1*2 = 2 2*1 = 2 2*2 = 4 which yields the set {1,2,4}. To circumvent this, we would need to replace multiplication with modular multiplication (see page 440 of Ryan (2000)). Hence, with the -1,+1 values for the main factors, we also have -1,+1 values for all interactions which in turn yields (for all terms) a consistent X of X = (+1) - (-1) = +2 In summary then, B = ( ) = ( Y) / 2 = (1/2) * ( Y) and so to achieve our goal of having the final coefficients reflect Y only, we simply gather up all of the 2's in the denominator and create a leading multiplicative constant of 1 with denominator 2, that is, 1/2. Example for k = 1 case For example, for the trivial k = 1 case, the obvious model Y = intercept + slope*X1 Y = c + ( )*X1 becomes Y = c + (1/ X) * ( Y)*X1 or simply Y = c + (1/2) * ( Y)*X1 Y = c + (1/2)*(factor 1 effect)*X1 Y = c + (1/2)*(B * )*X1, with B * = 2B = E This k = 1 factor result is easily seen to extend to the general k-factor case. 5.5.9.9.6. Motivation: Why is the 1/2 in the Model? http://www.itl.nist.gov/div898/handbook/pri/section5/pri5996.htm (4 of 4) [5/1/2006 10:31:36 AM] 5. Process Improvement 5.5. Advanced topics 5.5.9. An EDA approach to experimental design 5.5.9.9. Cumulative residual standard deviation plot 5.5.9.9.7.Motivation: What are the Advantages of the LinearCombinatoric Model? Advantages: perfect fit and comparable coefficients The linear model consisting of main effects and all interactions has two advantages: Perfect Fit: If we choose to include in the model all of the main effects and all interactions (of all orders), then the resulting least squares fitted model will have the property that the predicted values will be identical to the raw response values Y. We will illustrate this in the next section. 1. Comparable Coefficients: Since the model fit has been carried out in the coded factor (-1,+1) units rather than the units of the original factor (temperature, time, pressure, catalyst concentration, etc.), the factor coefficients immediately become comparable to one another, which serves as an immediate mechanism for the scale-free ranking of the relative importance of the factors. 2. Example To illustrate in detail the above latter point, suppose the (-1,+1) factor X1 is really a coding of temperature T with the original temperature ranging from 300 to 350 degrees and the (-1,+1) factor X2 is really a coding of time t with the original time ranging from 20 to 30 minutes. Given that, a linear model in the original temperature T and time t would yield coefficients whose magnitude depends on the magnitude of T (300 to 350) and t (20 to 30), and whose value would change if we decided to change the units of T (e.g., from Fahrenheit degrees to Celsius degrees) and t (e.g., from minutes to seconds). All of this is avoided by carrying out the fit not in the original units for T (300,350) and t (20,30), but in the coded units of X1 (-1,+1) and X2 (-1,+1). The resulting coefficients are unit-invariant, and thus the coefficient magnitudes reflect the true contribution of the factors and interactions without regard to the unit 5.5.9.9.7. Motivation: What are the Advantages of the LinearCombinatoric Model? http://www.itl.nist.gov/div898/handbook/pri/section5/pri5997.htm (1 of 2) [5/1/2006 10:31:36 AM] of measurement. Coding does not lead to loss of generality Such coding leads to no loss of generality since the coded factor may be expressed as a simple linear relation of the original factor (X1 to T, X2 to t). The unit-invariant coded coefficients may be easily transformed to unit-sensitive original coefficients if so desired. 5.5.9.9.7. Motivation: What are the Advantages of the LinearCombinatoric Model? http://www.itl.nist.gov/div898/handbook/pri/section5/pri5997.htm (2 of 2) [5/1/2006 10:31:36 AM] 5. Process Improvement 5.5. Advanced topics 5.5.9. An EDA approach to experimental design 5.5.9.9. Cumulative residual standard deviation plot 5.5.9.9.8.Motivation: How do we use the Model to Generate Predicted Values? Design matrix with response for 2 factors To illustrate the details as to how a model may be used for prediction, let us consider a simple case and generalize from it. Consider the simple Yates-order 2 2 full factorial design in X1 and X2, augmented with a response vector Y: X1 X2 Y - - 2 + - 4 - + 6 + + 8 Geometric representation This can be represented geometrically 5.5.9.9.8. Motivation: How do we use the Model to Generate Predicted Values? http://www.itl.nist.gov/div898/handbook/pri/section5/pri5998.htm (1 of 3) [5/1/2006 10:31:36 AM] Determining the prediction equation For this case, we might consider the model From the above diagram, we may deduce that the estimated factor effects are: c = = the average response = (2 + 4 + 6 + 8) / 4 = 5 B 1 = = average change in Y as X>1 goes from -1 to +1 ((4-2) + (8-6)) / 2 = (2 + 2) / 2 = 2 Note: the (4-2) is the change in Y (due to X1) on the lower axis; the (8-6) is the change in Y (due to X1) on the upper axis. B 2 = = average change in Y as X2 goes from -1 to +1 ((6-2) + (8-4)) / 2 = (4 + 4) / 2 = 4 B 12 = = interaction = (the less obvious) average change in Y as X1*X2 goes from -1 to +1 ((2-4) + (8-6)) / 2 = (-2 + 2) / 2 = 0 and so the fitted model (that is, the prediction equation) is or with the terms rearranged in descending order of importance Table of fitted values Substituting the values for the four design points into this equation yields the following fitted values X1 X2 Y - - 2 2 + - 4 4 - + 6 6 + + 8 8 Perfect fit This is a perfect-fit model. Such perfect-fit models will result anytime (in this orthogonal 2-level design family) we include all main effects and all interactions. Remarkably, this is true not only for k = 2 factors, but for general k. Residuals For a given model (any model), the difference between the response value Y and the predicted value is referred to as the "residual": residual = Y - The perfect-fit full-blown (all main factors and all interactions of all orders) models will have all residuals identically zero. The perfect fit is a mathematical property that comes if we choose to use the linear model with all possible terms. 5.5.9.9.8. Motivation: How do we use the Model to Generate Predicted Values? http://www.itl.nist.gov/div898/handbook/pri/section5/pri5998.htm (2 of 3) [5/1/2006 10:31:36 AM] Price for perfect fit What price is paid for this perfect fit? One price is that the variance of is increased unnecessarily. In addition, we have a non-parsimonious model. We must compute and carry the average and the coefficients of all main effects and all interactions. Including the average, there will in general be 2 k coefficients to fully describe the fitting of the n = 2 k points. This is very much akin to the Y = f(X) polynomial fitting of n distinct points. It is well known that this may be done "perfectly" by fitting a polynomial of degree n-1. It is comforting to know that such perfection is mathematically attainable, but in practice do we want to do this all the time or even anytime? The answer is generally "no" for two reasons: Noise: It is very common that the response data Y has noise (= error) in it. Do we want to go out of our way to fit such noise? Or do we want our model to filter out the noise and just fit the "signal"? For the latter, fewer coefficients may be in order, in the same spirit that we may forego a perfect-fitting (but jagged) 11-th degree polynomial to 12 data points, and opt out instead for an imperfect (but smoother) 3rd degree polynomial fit to the 12 points. 1. Parsimony: For full factorial designs, to fit the n = 2 k points we would need to compute 2 k coefficients. We gain information by noting the magnitude and sign of such coefficients, but numerically we have n data values Y as input and n coefficients B as output, and so no numerical reduction has been achieved. We have simply used one set of n numbers (the data) to obtain another set of n numbers (the coefficients). Not all of these coefficients will be equally important. At times that importance becomes clouded by the sheer volume of the n = 2 k coefficients. Parsimony suggests that our result should be simpler and more focused than our n starting points. Hence fewer retained coefficients are called for. 2. The net result is that in practice we almost always give up the perfect, but unwieldy, model for an imperfect, but parsimonious, model. Imperfect fit The above calculations illustrated the computation of predicted values for the full model. On the other hand, as discussed above, it will generally be convenient for signal or parsimony purposes to deliberately omit some unimportant factors. When the analyst chooses such a model, we note that the methodology for computing predicted values is precisely the same. In such a case, however, the resulting predicted values will in general not be identical to the original response values Y; that is, we no longer obtain a perfect fit. Thus, linear models that omit some terms will have virtually all non-zero residuals. 5.5.9.9.8. Motivation: How do we use the Model to Generate Predicted Values? http://www.itl.nist.gov/div898/handbook/pri/section5/pri5998.htm (3 of 3) [5/1/2006 10:31:36 AM] 5. Process Improvement 5.5. Advanced topics 5.5.9. An EDA approach to experimental design 5.5.9.9. Cumulative residual standard deviation plot 5.5.9.9.9.Motivation: How do we Use the Model Beyond the Data Domain? Interpolation and extrapolation The previous section illustrated how to compute predicted values at the points included in the design. One of the virtues of modeling is that the resulting prediction equation is not restricted to the design data points. From the prediction equation, predicted values can be computed elsewhere and anywhere: within the domain of the data (interpolation);1. outside of the domain of the data (extrapolation).2. In the hands of an expert scientist/engineer/analyst, the ability to predict elsewhere is extremely valuable. Based on the fitted model, we have the ability to compute predicted values for the response at a large number of internal and external points. Thus the analyst can go beyond the handful of factor combinations at hand and can get a feel (typically via subsequent contour plotting) as to what the nature of the entire response surface is. This added insight into the nature of the response is "free" and is an incredibly important benefit of the entire model-building exercise. Predict with caution Can we be fooled and misled by such a mathematical and computational exercise? After all, is not the only thing that is "real" the data, and everything else artificial? The answer is "yes", and so such interpolation/extrapolation is a double-edged sword that must be wielded with care. The best attitude, and especially for extrapolation, is that the derived conclusions must be viewed with extra caution. By construction, the recommended fitted models should be good at the design points. If the full-blown model were used, the fit will be perfect. If the full-blown model is reduced just a bit, then the fit will still typically be quite good. By continuity, one would expect perfection/goodness at the design points would lead to goodness in the immediate vicinity of the design points. However, such local goodness 5.5.9.9.9. Motivation: How do we Use the Model Beyond the Data Domain? http://www.itl.nist.gov/div898/handbook/pri/section5/pri5999.htm (1 of 2) [5/1/2006 10:31:36 AM] does not guarantee that the derived model will be good at some distance from the design points. Do confirmation runs Modeling and prediction allow us to go beyond the data to gain additional insights, but they must be done with great caution. Interpolation is generally safer than extrapolation, but mis-prediction, error, and misinterpretation are liable to occur in either case. The analyst should definitely perform the model-building process and enjoy the ability to predict elsewhere, but the analyst must always be prepared to validate the interpolated and extrapolated predictions by collection of additional real, confirmatory data. The general empirical model that we recommend knows "nothing" about the engineering, physics, or chemistry surrounding your particular measurement problem, and although the model is the best generic model available, it must nonetheless be confirmed by additional data. Such additional data can be obtained pre-experimentally or post-experimentally. If done pre-experimentally, a recommended procedure for checking the validity of the fitted model is to augment the usual 2 k or 2 k-p designs with additional points at the center of the design. This is discussed in the next section. Applies only for continuous factors Of course, all such discussion of interpolation and extrapolation makes sense only in the context of continuous ordinal factors such as temperature, time, pressure, size, etc. Interpolation and extrapolation make no sense for discrete non-ordinal factors such as supplier, operators, design types, etc. 5.5.9.9.9. Motivation: How do we Use the Model Beyond the Data Domain? http://www.itl.nist.gov/div898/handbook/pri/section5/pri5999.htm (2 of 2) [5/1/2006 10:31:36 AM] 5. Process Improvement 5.5. Advanced topics 5.5.9. An EDA approach to experimental design 5.5.9.9. Cumulative residual standard deviation plot 5.5.9.9.10.Motivation: What is the Best Confirmation Point for Interpolation? Augment via center point For the usual continuous factor case, the best (most efficient and highest leverage) additional model-validation point that may be added to a 2 k or 2 k-p design is at the center point. This center point augmentation "costs" the experimentalist only one additional run. Example For example, for the k = 2 factor (Temperature (300 to 350), and time (20 to 30)) experiment discussed in the previous sections, the usual 4-run 2 2 full factorial design may be replaced by the following 5-run 2 2 full factorial design with a center point. X1 X2 Y - - 2 + - 4 - + 6 + + 8 0 0 Predicted value for the center point Since "-" stands for -1 and "+" stands for +1, it is natural to code the center point as (0,0). Using the recommended model we can substitute 0 for X1 and X2 to generate the predicted value of 5 for the confirmatory run. 5.5.9.9.10. Motivation: What is the Best Confirmation Point for Interpolation? http://www.itl.nist.gov/div898/handbook/pri/section5/pri599a.htm (1 of 2) [5/1/2006 10:31:37 AM] [...]... for assessing general model adequacy http://www.itl.nist.gov/div898/handbook/pri/section5/pri599a.htm (2 of 2) [5/1/2006 10:31:37 AM] 5.5.9.9.11 Motivation: How do we Use the Model for Interpolation? 5 Process Improvement 5.5 Advanced topics 5.5.9 An EDA approach to experimental design 5.5.9.9 Cumulative residual standard deviation plot 5.5.9.9.11 Motivation: How do we Use the Model for Interpolation?... value for interpolated data point Thus http://www.itl.nist.gov/div898/handbook/pri/section5/pri599b.htm (3 of 3) [5/1/2006 10:31:37 AM] 5.5.9.9.12 Motivation: How do we Use the Model for Extrapolation? 5 Process Improvement 5.5 Advanced topics 5.5.9 An EDA approach to experimental design 5.5.9.9 Cumulative residual standard deviation plot 5.5.9.9.12 Motivation: How do we Use the Model for Extrapolation?... plots and can add significant insight and understanding as to the nature of the response surface relating Y to the X's But, again, a final word of caution: the "pseudo data" that results from the modeling process is exactly that, pseudo-data It is not real data, and so the model and the model's predicted values must be validated by additional confirmatory (real) data points A more balanced approach is that:... as an easy-to-interpret tool for determining a good and parsimonious model http://www.itl.nist.gov/div898/handbook/pri/section5/pri599c.htm (2 of 2) [5/1/2006 10:31:38 AM] 5.5.9.10 DEX contour plot 5 Process Improvement 5.5 Advanced topics 5.5.9 An EDA approach to experimental design 5.5.9.10 DEX contour plot Purpose The dex contour plot answers the question: Where else could we have run the experiment... effect of factor X1 will change depending on the setting of factor X3 http://www.itl.nist.gov/div898/handbook/pri/section5/pri59a.htm (4 of 4) [5/1/2006 10:31:38 AM] 5.5.9.10.1 How to Interpret: Axes 5 Process Improvement 5.5 Advanced topics 5.5.9 An EDA approach to experimental design 5.5.9.10 DEX contour plot 5.5.9.10.1 How to Interpret: Axes What factors go on the 2 axes? For this first item, we choose... axis: component 2 from the item 1 interaction (e.g., X4) http://www.itl.nist.gov/div898/handbook/pri/section5/pri59a1.htm (3 of 3) [5/1/2006 10:31:38 AM] 5.5.9.10.2 How to Interpret: Contour Curves 5 Process Improvement 5.5 Advanced topics 5.5.9 An EDA approach to experimental design 5.5.9.10 DEX contour plot 5.5.9.10.2 How to Interpret: Contour Curves Non-linear appearance of contour curves implies... the contour plot implied a strong X1*X3 interaction http://www.itl.nist.gov/div898/handbook/pri/section5/pri59a2.htm (2 of 2) [5/1/2006 10:31:39 AM] 5.5.9.10.3 How to Interpret: Optimal Response Value 5 Process Improvement 5.5 Advanced topics 5.5.9 An EDA approach to experimental design 5.5.9.10 DEX contour plot 5.5.9.10.3 How to Interpret: Optimal Response Value Need to define "best" We need to identify... are trying to maximize the response, the selected optimal value is 100 http://www.itl.nist.gov/div898/handbook/pri/section5/pri59a3.htm [5/1/2006 10:31:39 AM] 5.5.9.10.4 How to Interpret: Best Corner 5 Process Improvement 5.5 Advanced topics 5.5.9 An EDA approach to experimental design 5.5.9.10 DEX contour plot 5.5.9.10.4 How to Interpret: Best Corner Four corners representing 2 levels for 2 factors... defective springs data the best corner is (+,+) http://www.itl.nist.gov/div898/handbook/pri/section5/pri59a4.htm (2 of 2) [5/1/2006 10:31:39 AM] 5.5.9.10.5 How to Interpret: Steepest Ascent/Descent 5 Process Improvement 5.5 Advanced topics 5.5.9 An EDA approach to experimental design 5.5.9.10 DEX contour plot 5.5.9.10.5 How to Interpret: Steepest Ascent/Descent Start at optimum corner point From the . or complicated the model is, provided that the design is orthogonal. This process greatly simplifies the model-building process because the need to recalculate all of the model coefficients for. Model? http://www.itl.nist.gov/div898/handbook/pri/section5/pri5996.htm (4 of 4) [5/1/2006 10:31:36 AM] 5. Process Improvement 5.5. Advanced topics 5.5.9. An EDA approach to experimental design 5.5.9.9. Cumulative. Model? http://www.itl.nist.gov/div898/handbook/pri/section5/pri5997.htm (2 of 2) [5/1/2006 10:31:36 AM] 5. Process Improvement 5.5. Advanced topics 5.5.9. An EDA approach to experimental design 5.5.9.9. Cumulative